Systematic Listing and Product Rule –

Systematic listing and the product rule for counting are tested on GCSE Maths papers across AQA, Edexcel and OCR at Foundation and Higher tier. These skills ensure you can list every possible outcome of a combined event without missing any or repeating them, and quickly calculate the total number of outcomes without writing them all out. Exam questions range from simple menu combinations to digit arrangements and lock codes. This guide covers both skills with clear examples and tells you exactly when to use each method. For the full specification overview, see our complete GCSE Maths topics list.

What Is Systematic Listing?

Systematic listing means writing out all possible outcomes in an organised way so that none are missed and none are repeated. You fix one element and vary the others methodically.

The Product Rule for Counting

The product rule states that if there are m ways of doing one thing and n ways of doing another, the total number of combined outcomes is:

This extends to three or more events: if there are m ways, then n ways, then p ways, the total is m × n × p.

Key Formulas

For example, a 3-course meal with 4 starters, 3 mains and 5 desserts gives 4 × 3 × 5 = 60 different meals.

Step-by-Step Method

Systematic Listing

1. Identify the categories (e.g. starter, main, dessert or digit 1, digit 2).
2. Fix the first category at its first option.
3. List all combinations for the remaining categories.
4. Move to the next option for the first category and repeat.
5. Continue until all options for the first category are exhausted.
6. Count the total and check it matches the product rule.

Using the Product Rule

1. Count the number of choices for each stage or event.
2. Multiply them together.
3. If there are restrictions (e.g. "no repeats"), adjust the count at each stage accordingly.

Worked Example 1 — Foundation Level

Question: A cafe offers 3 sandwiches (ham, cheese, tuna) and 4 drinks (tea, coffee, juice, water). How many different lunch combinations of one sandwich and one drink are possible?

Working:

Using the product rule: 3 × 4 = 12.

Systematic list: Ham-Tea, Ham-Coffee, Ham-Juice, Ham-Water, Cheese-Tea, Cheese-Coffee, Cheese-Juice, Cheese-Water, Tuna-Tea, Tuna-Coffee, Tuna-Juice, Tuna-Water.

Answer: 12 different lunch combinations.

Worked Example 2 — Higher Level

Question: A 3-digit code is formed using the digits 1, 2, 3, 4, 5. Each digit may only be used once. How many different codes are possible?

Working:

First digit: 5 choices. Second digit: 4 remaining choices. Third digit: 3 remaining choices.

Total = 5 × 4 × 3 = 60.

Answer: 60 different codes.

Worked Example 3 — Exam Style

Question: Amy has 3 skirts (black, grey, navy), 4 tops (red, white, blue, green) and 2 pairs of shoes (trainers, boots). (a) How many different outfits can she make? (b) List all outfits that include the black skirt and trainers.

Working:

(a) Product rule: 3 × 4 × 2 = 24 outfits.

(b) Fix black skirt and trainers. Only the top varies: Black-Red-Trainers, Black-White-Trainers, Black-Blue-Trainers, Black-Green-Trainers.

Answer: (a) 24 outfits. (b) 4 outfits: Black-Red-Trainers, Black-White-Trainers, Black-Blue-Trainers, Black-Green-Trainers.

Common Mistakes

• Missing combinations when listing. Without a systematic approach, students skip outcomes. Always fix one category and cycle through the others.
• Forgetting "without replacement" restrictions. If digits or items cannot be repeated, reduce the number of choices at each stage.
• Using the product rule when order does not matter. If the question asks for combinations (where order is irrelevant), the product rule overcounts — you may need to divide by the number of arrangements. At GCSE, most questions treat order as mattering.

Exam Tips

• If the question says "list all possibilities", you must write every outcome — the product rule alone will not earn full marks.
• If it says "how many", use the product rule for speed and show your multiplication.
• When digits cannot repeat, think of it as choosing for each position in turn: the number of choices decreases by 1 at each step.
• Check your list total matches the product rule to confirm you have not missed any.
• For related probability, see sample space diagrams and combined events probability.

Practice Questions

Q1 (Foundation): A pizza restaurant offers 3 bases (thin, thick, stuffed) and 5 toppings. How many different single-topping pizzas are possible?

Q2 (Foundation): List all the two-letter codes that can be made from the letters A, B and C if each letter can be used only once.

Q3 (Higher): A 4-digit PIN is formed using digits 0–9 with repetition allowed. How many PINs are possible? How many PINs start with an odd digit?

Practise systematic listing and the product rule for free on GCSEMathsAI.

Related Topics

• Sample Space Diagrams
• Combined Events Probability
• Probability Basics and Relative Frequency

Summary

Systematic listing means writing out all possible outcomes in an organised way by fixing one element and varying the rest. The product rule for counting gives the total number of outcomes as the product of the choices at each stage: m × n × p × ... If items cannot be repeated, reduce the number of choices at each subsequent stage. Use the product rule for "how many" questions and systematic listing when the question asks you to show all possibilities. Always cross-check your list count against the product rule to make sure nothing is missing.